A good book for metric spaces?

Question

I'm looking for some book to study metric spaces. 2 years ago, used a book called Burkill, as well as using multiple topological concepts, I have also studied the Munkres, Chapters 2,3,4,5,6,9. Anyway, I do not feel as comfortable as well in group theory, rings or general topology )=. So I feel that books such as Hungerford, Robinson Group, Atiyah Mcdonald. They have been very important in my formation. But I have not had that feeling with any book of analysis (metric spaces), I feel my training is still very basic. What book would you recommend to study analysis?

Answer 1

A good book for metric spaces specifically would be Ó Searcóid's Metric Spaces. However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis.

A good book for real analysis would be Kolmogorov and Fomin's Introductory Real Analysis.

Answer 2

There is a wonderful short book by Kaplansky called Set Theory and Metric Spaces.

How good is it? I was able to read it at the beginning of my undergraduate career, and in the intervening years of undergraduate study, graduate study and then mathematical research, I have only ever turned to other books on either of these subjects out of idle curiosity: everything I have ever needed to know is in Kaplansky's text.

Answer 3

Rudin, Principles of Mathematical Analysis.

Answer 4

Good introduction to Metric Spaces Metric Spaces: Iteration and Application

Answer 5

Dieudonné's Foundations of modern analysis Chapter 3 is a very thorough treatment of metric spaces. It's a bit dry, but perfect as reference.